We say that a function \(f\)  is the optimal \(\chi \) -binding function for a \(\chi \) -bounded class of graphs \(\mathcal {G}\)  if for every positive integer \(t\) , there is a graph \(G\in \mathcal {G}\)  such that \(\omega (G)=t\)  and \(\chi (G)=f(t)\) . For example, \(f(x)=x\)  is the optimal \(\chi \) -binding function for the class of perfect graphs. Likewise, \(f(x)=\lceil \frac{5x}{4}\rceil \)  is the optimal \(\chi \) -binding function for the class of (\(P_6, C_4\) )-free graphs, (\(P_5, P_4\vee K_1\) )-free graphs and (\(P_4+K_1, P_4\vee K_1\) )-free graphs, since certain graph in such a class can be constructed from a \(C_5\)  by `blowing up' each vertex into a clique; see [1]}. However, in general, the problem of finding the optimal \(\chi \) -binding function for a given \(\chi \) -bounded class of graphs often seems to be difficult. So it is interesting to study such graphs with small clique number in order to obtain the optimal \(\chi \) -binding function.
It is worth mention here that the problem of finding the chromatic number of graphs with small clique number have been motivating
research in graph
theory and has a long history starting from the `Four color problem'. Recent surveys and literature show that it still receives special attention. A constructive proof of Mycielski [2]} says that for any positive integer \(k\) , there is a graph with chromatic number \(k\)  and clique number 2.
We give below few examples for the chromatic number of graphs (in particular, graphs defined by some forbidden induced subgraphs) with small clique number. We refer to the recent surveys [3]}, [4]} for more details and references.
